Download Chapter 7 Quantum Theory and the Electronic Structure of Atoms

Chapter 7
Quantum Theory and the Electronic Structure of Atoms
This chapter introduces the student to quantum theory and the importance of this theory in describing
electronic behavior. Upon completion of Chapter 7, the student should be able to:
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Explain how Planck’s theory challenged classical physics.
Define wavelength, frequency, and amplitude of waves.
Utilize the relationship between speed, wavelength, and frequency (hertz).
Describe Maxwell’s theory of electromagnetic radiation.
Recall from memory the speed of light (3.00 x 108 m/s).
Apply the metric unit of nano in calculations involving wavelength of light.
Classify various regions of the electromagnetic spectrum in terms of energy, frequency and
wavelength.
Use Planck’s equation to determine energy, frequency, or wavelength of electromagnetic radiation.
Describe the photoelectric effect as explained by Einstein using such terms as threshold frequency,
photons, kinetic energy, binding energy, light intensity and number of electrons emitted.
Show how Bohr’s model of the atom explains emission, absorption and line spectra for the hydrogen
atom.
Compare Bohr’s model of the atom and that of the sun and surrounding planets.
Predict the wavelength (frequency) of electromagnetic radiation emitted (absorbed) for electronic
transitions in a hydrogen atom.
Use the terms ground state and excited state to describe electronic transitions.
Describe De Broglie’s relationship involving the wavelength of particles.
Explain why for common objects traveling at reasonable speeds the corresponding wavelength
becomes vanishingly small.
Explain the major components of a laser and list three properties that are characteristic of a laser.
Describe Heisenberg’s uncertainty principle.
Contrast orbits (shells) in Bohr’s theory with orbitals in quantum theory.
Discuss the concept of electron density.
Recall from memory the four quantum numbers (n, ℓ, mℓ, ms) and their relationships.
Relate the values of the angular momentum quantum number, ℓ, to common names for each orbital
(s, p, d, f) and describe their shapes.
Account for the number of orbitals and number of electrons associated with each value of ℓ, the
angular momentum quantum number.
Categorize orbital energy levels in many-electron atoms in order of increasing energy.
Write the four quantum numbers for all electrons in multi-electron atoms.
Predict the electron configuration and orbital diagrams for multi-electron atoms using the Pauli
exclusion principle and Hund’s rule.
Deduce orbital diagrams from diamagnetic and paramagnetic data.
Derive the ground state electron configuration of multi-electron atoms using the Aufbau principle.
List several exceptions to the expected electron configuration for common metals (Cr, Mo, Cu, Ag,
and Au).
Section 7.1
From Classical Physics to Quantum Theory
Max Planck started the “revolution” that suggests that classical physics does not govern atoms and
molecules. He suggested that energy comes in discrete quantities, or quanta. Planck’s theory requires an
understanding of waves which include wavelength, frequency and amplitude. The speed of any wave is the
product of its wavelength and its frequency. The speed of electromagnetic radiation is 3.00 × 108 meters per
second or 186,000 miles per second. Your author introduces the concept of lasers in the Chemistry in Action
featuring “Laser-the Splendid Light”. One of the principles of lasers is that light bounces back and forth
between mirrors inside the laser cavity. Since light travels at about 186,000 miles per second, it will travel
about one foot in 1 × 10-9 seconds. With this in mind, it is easy to see that light makes a huge number of
passes between the mirrors in a typical laser cavity in a very short period of time.
Students find it interesting to determine the wavelength of their favorite radio station. For example, if a
local station broadcasts at 109.5 megahertz, the wavelength can be determined.
c = λν
 1x106 / s 
3.00 x 108 m/s = (λ )(109.5 MHz) 

 1 MHz 


λ=
3.00 x 108
109.5 x 106
m
λ = 2.74 m
This is in contrast to green light which has a wavelength of 522 nm.
A direct result of Planck’s theory is that energy emitted or absorbed is directly proportional to frequency.
E = hν
Planck’s constant, h, has the value of 6.63 × 10-34 J⋅s. Sometimes it is easier for students to use Planck’s
relationship, if we think of Planck’s constant having the units of J⋅s/object where the term “object” may refer to
photon or electron. For example, how much energy is emitted by 1.00 mole of photons having the frequency
equal to 5.75 × 1014 Hz?
14


-34 J • s  5.75 x 10
E = hν =  6.63 x 10

photon  
s





E = 3.81 x 10-19 J / photon
Each photon carries 3.81 × 10-19 J; therefore,
(3.81 x 10-19 J / photon)(6.022 x 1023 photon / mol photon) = 2.29 x 105 J/mol photon
a mole of photons would represent 2.29 × 105 Joules.
Section 7.2
The Photoelectric Effect
The photoelectric effect, the generation of electricity by converting radiant energy to electrical energy, is
demonstrated by light sensors. An industrial application of this effect is testing light sources used in the
ultraviolet curing of polymers. Since no electrons are ejected below the threshold frequency, light sensors
need to be selected for specific wavelengths of light being examined. For example, if one wishes to
determine the intensity of a visible light source, it would be unwise to use an ultraviolet light sensor. This is
because the threshold frequency for the ultraviolet sensor is higher than the frequency of visible light;
therefore, no matter how intense the visible light is, it will not register on the ultraviolet sensor.
Section 7.3
Bohr’s Theory of the Hydrogen Atom
It is interesting to note that astronomy had a considerable influence on Bohr’s model of the atom. The
logic used was if the planets revolve around the sun in specific orbits, then it would be reasonable that the
electrons would revolve about the nucleus in specific orbits.
Students often wonder why we would even consider Bohr’s theory when it explains only the hydrogen
atom. They should be reassured that even though the theory is limited, it is useful in gaining the foundation
required to understand more complex theories. Bohr’s theory introduces the concepts of ground state, excited
state, absorption and emission, for example.
An extension to Bohr’s theory is using Rydberg’s constant (2.18 × 10-18 J) and the equation
 1 1 
∆E = RH  - 
 n2 n2 
f
 i
to determine the energy that is absorbed or emitted by specific transitions. Once the energy is determined,
both the frequency and wavelength can be determined using Planck’s relationship.
Section 7.4
The Dual Nature of the Electron
Louis de Broglie suggested that if light can have both wave nature and particle nature, then electrons
can behave like waves even though they have a known mass. De Broglie’s relation is
λ=
h
mv
Students have difficulty accepting that particles can have wave nature and light can have particle nature.
They should be reassured that their misgivings are founded on the fact that they have never “experienced”
particles as small as electrons; therefore, they have to accept this concept as reasonable based on
experimental evidence. The evidence is demonstrated by your author in the Chemistry in Action essay
“Electron Microscopy".
Section 7.5
Quantum Mechanics
Bohr’s theory is successful in describing the spectra for hydrogen atoms, but falls short for atoms with
more than one electron. These shortcomings gave rise to development of the quantum mechanical
description of atoms as suggested by Erwin Schrödinger. Quantum mechanics describe electron density
which is the probability that an electron will be found in a particular region. This is a direct application of
Werner Heisenberg’s uncertainty principle, which suggests that it is not possible to know the exact position
and the momentum of a particle simultaneously. Therefore, the concept of defined orbits for electrons yields
to using atomic orbitals.
Section 7.6
Quantum Numbers
Our students understand that the equation y = mx + b describes a straight line. They know that m is the slope
and b is the y- intercept. The quantum numbers are similar in that they are the mathematical solution to the
Schrödinger equation. The principal quantum number, n, gives an indication of the total energy that an
electron possesses. In a general way, it describes the average distance the electron is from the nucleus. The
largest value n can be for a ground state electron is seven. The angular momentum quantum number, ℓ,
describes the shape of the orbital. The magnetic quantum number, mℓ, sets the direction in space for each
orbital. The electron spin quantum number, ms, suggests that each electron either spins clockwise or counterclockwise. The rules for the possible values for each of the four quantum numbers are:
n = 1, 2, 3…
ℓ = 0, 1, 2, 3…, n - 1
mℓ = --ℓ…0…+ℓ
ms = + 1/2 or – 1/2
The larger the value of n, the further the electron is from the nucleus. Only positive integers are possible for
n. The angular momentum quantum number starts with the value of zero and can be any integer up to and
including one less than n. That is to say that if n equals five, then ℓ can be zero, one, two, three or four. It
cannot be five or greater. The magnetic quantum number, mℓ, can be negative as well as positive. The
possible values are from minus ℓ to zero to positive ℓ. Therefore, if ℓ is two, then mℓ can be minus two, minus
one, zero, plus one and plus two. The electron spin quantum number can have the values of plus one-half or
minus one-half. These two values correspond to spin up or spin down, as written by arrows, or clockwise and
counter-clockwise.
Section 7.7
Atomic orbitals
In the previous section, we describe the four quantum numbers. In this section your author describes
the shape of each of the orbitals. If ℓ equals zero (s orbital), then the shape is spherical. The probability of
finding an electron is within a certain shell at a given distance from the nucleus. If ℓ is one (p orbital), then the
shape of the orbital is much like a three-dimensional figure eight with the cross overpoint being the nucleus.
When ℓ is two (d orbitals), the shape of the orbitals are two-fold. Four of the orbitals are like threedimensional four-leaf clovers and the fifth is the shape of a three-dimensional figure eight with a doughnut
surrounding its mid-section. See Figures 7.19, 7.20, and 7.21. The angular momentum quantum number, mℓ,
describes how each of the above orbitals are oriented in space.
For hydrogen, a single-electron atom, all orbitals with the same principle quantum are the same energy.
That is to say for a given value of n, s, p, and d are all the same energy. This is not true for multiple electron
atoms where the s orbital is lower energy than p orbitals, which are lower energy than d orbitals for a given
value of n.
Section 7.8
Electron Configuration
There are three common ways to describe each electron in an atom. The first is to use the notation
(n, ℓ, mℓ, ms)
where n, ℓ, mℓ, and ms have been described above. A second method is to use electron configuration. This is
described by the example
2s1
where the two corresponds to the principal quantum number of two, the s represents the s orbital or ℓ of zero
and the superscript one suggests that there is one electron with this value of ℓ. The third method is the “box
diagram”. This is represented by
↑
2s
where the box is labeled as 2s and has one electron in that orbital with a plus spin. It should be noted that
each box represents a single orbital. Each orbital can have a maximum of two electrons (spin up or spin
down). Thus when ℓ equals one, there are three boxes (three values of mℓ) and they could be represented by
2p
An interesting question for your students is to ask them to indicate what the maximum number of electrons a
3p orbital can have. They may very well indicate six, but the correct answer is two. (That is the maximum for
all orbitals). The logic of answering six is that there are six electrons that can be labeled as 3p electrons.
However, these six electrons occupy three orbitals with two electrons per orbital.
Pauli exclusion principle states that no two electrons within a given atom can have four identical
quantum numbers. It is important to stress that this is for a given atom, because as we will see in the next
section, an electron configuration for lithium uses the descriptions for the two electrons from helium when
describing two of the three electrons.
It is useful to introduce the concepts of paramagnetism and diamagnetism. Paramagnetic materials
have unpaired electrons. They have one or more orbitals (energy box) with a single electron. Diamagnetic
materials have all orbitals containing two electrons (each energy box contains two electrons, one with a plus
one-half spin and the other with a minus one-half spin). Paramagnetic materials are attracted by a magnet
while diamagnetic materials are slightly repelled by a magnet. Since this is an experimental property, it can
be used to justify Hund’s rule. Hund’s rule states that the most stable electron arrangement of electrons is the
one which has the greatest number of parallel spins. For example, carbon has six electrons to account for.
The first two electrons (lowest energy electrons) fill the energy box labeled 1s.
↑↓
1s
The next two go into the next lowest energy level which is 2s.
↑↓
2s
We know that there are two electrons remaining to be described. Experimentally, we can determine that
carbon is paramagnetic, thus it must have unpaired electrons. It follows then that the energy diagram for the
2p electrons must be
↑
2p
↑
and not
2p
↑↓
which would result in a diamagnetic material. With this in mind, it is apparent that the energy box diagram
gives a bit of information that is not shown by standard electron configuration. For example, the electron
configuration of oxygen is
1s 2 2 s 2 2 p 4
The energy box diagram would be
↑↓
↑
↑↓
2s
↑↓
1s
↑
2p
which gives information about the paramagnetic properties of oxygen atoms. The electron configuration of
oxygen could easily give the misconception that all electrons in oxygen are paired, but they are not.
Section 7.9
The Building-up Principle
The Aufbau principle (building-up principle) describes how we can “build” the periodic table. In fact, we
can use the periodic table to assist us in describing the correct electron configuration of each of the elements
(certainly there are many exceptions; however, most common elements can be described by the following
method).
First, we always use the previous noble gas as our core. For lithium the previous noble gas is He and
for sodium the previous noble gas is Ne, etc. The core noble gas is placed in square brackets and the rest of
the configuration follows. For example, sodium’s electron configuration is
[Ne]3s1
[Ne] core accounts for ten electrons and the 3s1 gives us the eleventh electron. There are five exceptions that
should be noted because these elements correspond to metals that are commonly used in industry. The
expected electron configuration for Cr (Z = 24) is [Ar] 4s23d4 but the experimentally determined configuration
is [Ar]4s13d5. In a similar fashion, for Mo (Z = 42) the correct electron configuration is [Kr]5s14d5 and not
[Kr]5s24d4. There are three elements commonly known as the coinage metals (Cu, Ag, and Au which have
traditionally been used as metals for coins). These three also have unexpected electron configurations. Cu
has [Ar]4s13d10 (not [Ar]4s23d9), Ag has [Kr]5s14d10 (not [Kr]5s24d9) and Au has [Xe]6s14f145d10 (not
[Xe]6s24f145d9). Table 7.3 gives the complete set of ground-state electron configurations. There are many
other “exceptions”; however, it is not worth confusing the students about these details.